Going up, going down

The idea

Linear growth is a linear pattern in which a quantity increases by a constant amount over equal intervals; linear decay is one in which it decreases by a constant amount over equal intervals.

A taxi and a tank

A taxi trip costs ₹100 to start and ₹60 for every km. A water tank is 3 m deep at the start of summer and the water sinks 0.5 m every month.

As the km and the months go by, which of these goes up and which goes down?

• The cost goes up, the water goes down
• Both go up
• Both go down

Watch both change

First the taxi. Press the button to travel one more km and watch the cost.

Now the tank. Press the button for one more month.

Growth and decay

Linear growth is a linear pattern in which a quantity increases by a constant amount over equal intervals. Linear decay is one in which it decreases by a constant amount over equal intervals.

• Situation | Change each step
• Taxi: C(d) = 100 + 60d | +₹60 per km (growth)
• Tank: h(t) = 3 − 0.5t | −0.5 m per month (decay)

Notes

Linear growth: a quantity increases by a constant amount over equal intervals. Linear decay: it decreases by a constant amount over equal intervals.

Check yourself

Bela has ₹100 and spends ₹5 every day. The money she has left shows linear…

The taxi cost is C(d) = 100 + 60d. What is the cost for 10 km?

Answer: 700

100 + 60 × 10 = 100 + 600 = 700 rupees.

A plant is 10 cm tall and grows 2 cm every week. This is linear…

The tank height is h(t) = 3 − 0.5t. What is the height after 4 months?

Answer: 1

3 − 0.5 × 4 = 3 − 2 = 1 metre.

• decay — correct. Yes. It goes down by the same ₹5 every day.
• growth. The amount goes down, so it is decay.
• decay. The height goes up, so it is growth.
• growth — correct. Yes. It goes up by the same 2 cm every week.